
theorem :: THEOREM 4.18. (2) iff (7)
  for L be Boolean LATTICE holds L is arithmetic iff ( L is complete &
  for x be Element of L ex X be Subset of L st X c= ATOM L & x = sup X )
proof
  let L be Boolean LATTICE;
  hereby
    assume
A1: L is arithmetic;
    then L opp is continuous by Th9,YELLOW_7:38;
    then L is completely-distributive by A1,WAYBEL_6:39;
    hence L is complete & for x be Element of L ex X be Subset of L st X c=
    ATOM L & x = sup X by Lm5;
  end;
  assume L is complete & for x be Element of L ex X be Subset of L st X c=
  ATOM L & x = sup X;
  then ex X be set st L,BoolePoset X are_isomorphic by Lm6;
  hence thesis by Th10,WAYBEL_1:6;
end;
